You can calculate the nth term in an arithmetic sequence using the following formula:

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Oct 27, 2012

arithmetic and geometric seriesby: Staff

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Part II

an = a₁ + (n - 1) * (d)

[An arithmetic sequence has the same form as a linear function: y = mx + b. However, the domain (the set of all input values; in this case n - 1) for an arithmetic sequence is limited to positive integers.]

an = a with a subscript of n (this is the nth term in the series)

a₁ = a with a subscript of 1 (this is the 1st term in the series)

n = number of terms

d = difference between consecutive terms (the common difference)

d, the common difference, can be calculated as follows:
d = an - an-1 (n must be greater than 1)

An “arithmetic series” is the “sum of an arithmetic sequence”. It is “n” multiplied by the average (arithmetic mean) of the first and last terms. [A sequence is an ordered list of numbers such as 1, 2, 3, 4. The sum of the terms of a sequence is called a series (for example, 1+2+3+4 is a series).]

Sn = (n/2)*(a1 + an)

or

Sn = (n/2)*[2a1 + (n – 1)d]

An arithmetic progression is a “divergent series”. Therefore, the sum of an infinite arithmetic progression cannot be calculated.

What an geometric sequence (geometric progression) looks like .

A geometric sequence has the (general) form:

an = a1 * (r)(n - 1)

an = a with a subscript of n is the nth term in the sequence

a1 = a with a subscript of 1 is the 1st term in the sequence

n = number of terms

r = the common ratio

r, the common ratio, can be calculated as follows:
r = an/an-1 (n must be greater than 1)

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Oct 27, 2012

arithmetic and geometric seriesby: Staff

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Part III

A “geometric series” is the “sum of an geometric sequence”. [A sequence is an ordered list of numbers such as 1, 2, 4, 8. The sum of the terms of a sequence is called a series (for example, 1+2+4+8 is a series).]

Sn = a1(1 – rn) / (1 – r)

S∞ = a1 / (1 – r)

This formula is only valid when -1< r < 1 (“convergent series”).

1.) -2/5 + (-3/20) + 1/10 + 7/20 + ..... + 8/5
.

This is an arithmetic series because the difference between each of the terms is a constant, d:

d, the common difference = ¼

(-3/20) - (-2/5) = ¼
(1/10) - (-3/20) = ¼
(7/20) - (1/10) = ¼

To find the sum of the terms, you must first determine how many terms are in the sequence:
-2/5, -3/20, 1/10, 7/20, ....., 8/5

Use the following formula. Substitute 8/5 for an. Substitute -2/5 for a1. Substitute ¼ for d. Solve for “n”.

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Oct 27, 2012

arithmetic and geometric seriesby: Staff

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Part IV

an = a1 + (n - 1) * (d)

8/5 = -2/5 + (n - 1) * (1/4)

8/5 + 2/5 = -2/5 + (n - 1) * (1/4) + 2/5

10/5 = -2/5 + (n - 1) * (1/4) + 2/5

2 = -2/5 + (n - 1) * (1/4) + 2/5

2 = -2/5 + 2/5 + (n - 1) * (1/4)

2 = 0 + (n - 1) * (1/4)

2 = (n - 1) * (1/4)

2 * 4 = (n - 1) * (1/4) * 4

8 = (n - 1) * (1/4) * 4

8 = (n - 1) * (4/4)

8 = (n - 1) * 1

8 = (n - 1)

8 = n - 1

8 + 1 = n - 1 + 1

9 = n - 1 + 1

9 = n + 0

9 = n

n = 9, there are 9 terms in the series

To find the sum of the series, use the following formula. Substitute 9 for “n”. Substitute ¼ for “d”. Substitute -2/5 for a1.